Questions D2 (553 questions)

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Edexcel D2 2017 June Q2
10 marks Moderate -0.3
2. The table shows the cost, in pounds, of transporting one unit of stock from each of three supply points, \(\mathrm { A } , \mathrm { B }\) and C , to each of four demand points, \(1,2,3\) and 4 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required.
1234Supply
A1517201133
B1211182121
C1813101625
Demand21172813
  1. Use the north-west corner method to obtain an initial solution.
    (1)
  2. Taking A4 as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
    (2)
  3. Taking the most negative improvement index to indicate the entering cell, use the stepping-stone method once to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.
  4. Determine whether your current solution is optimal, giving a reason for your answer.
Edexcel D2 2017 June Q3
13 marks Standard +0.8
3. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3
A plays 10- 26
A plays 2341
A plays 3- 11- 3
  1. Identify the play safe strategies for each player.
  2. State, giving a reason, whether there is a stable solution to this game.
  3. Find the best strategy for player A.
  4. Find the value of the game to player B.
Edexcel D2 2017 June Q4
7 marks Moderate -0.8
4. Four workers, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , are to be assigned to four tasks, \(1,2,3\) and 4 . Each worker must be assigned to only one task and each task must be done by only one worker. Worker A cannot do task 3 and worker D cannot do task 2
The cost, in pounds, of assigning each worker to each task is shown in the table below.
1234
A5384-20
B87724138
C70515225
D45-8170
The total cost is to be minimised.
Formulate the above situation as a linear programming problem. You must define your decision variables and make the objective function and constraints clear. You do not need to solve this problem.
Edexcel D2 2017 June Q5
13 marks Standard +0.3
5. The tableau below is the initial tableau for a three-variable linear programming problem in \(x , y\) and \(z\). The objective is to maximise the profit, \(P\).
Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)15- 23100180
\(s\)101101080
\(t\)16- 2001100
\(P\)- 1- 2- 50000
  1. Using the information in the tableau, write down
    1. the objective function,
    2. the three constraints as inequalities.
  2. Taking the most negative number in the profit row to indicate the pivot column at each stage, solve this linear programming problem. Make your method clear by stating the row operations you use.
  3. State the final values of the objective function and each variable.
Edexcel D2 2017 June Q6
12 marks Moderate -0.5
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d5798c81-290a-4e4b-aa46-497b62ca899b-07_1155_1541_223_264} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network. The number on each arc represents the capacity of the corresponding arc. The numbers in circles represent an initial flow from S to T .
  1. State the value of the initial flow.
  2. State the capacity of cut \(C _ { 1 }\)
  3. Complete the initialisation of the labelling procedure on Diagram 1 in the answer book by entering values along \(\mathrm { AC } , \mathrm { SB } , \mathrm { BE } , \mathrm { DE }\) and FG .
    (2)
  4. Hence use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  5. Draw a maximal flow pattern on Diagram 2 in the answer book.
  6. Prove that your flow is maximal.
Edexcel D2 2017 June Q7
13 marks Moderate -0.3
7. A clothing manufacturer can export a maximum of five batches of shirts each year. Each exported batch contains just one type of shirt, the types being T-shirts, Rugby shirts and Polo shirts. The table below shows the profit, in £ 1000 s, for the number of each exported batch type.
ABCDEF
A-8375826997
B83-9410377109
C7594-97120115
D8210397-105125
E6977120105-88
F9710911512588-
2.
1234Supply
A1517201133
B1211182121
C1813101625
Demand21172813
  1. 1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    1234Supply
    A33
    B21
    C25
    Demand21172813
    3.
    B plays 1B plays 2B plays 3
    A plays 10- 26
    A plays 2341
    A plays 3- 11- 3
    4.
    1234
    A5384-20
    B87724138
    C70515225
    D45-8170
    5. (a)
  2. You may not need to use all of these tableaux
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)15- 23100180
    \(s\)101101080
    \(t\)16- 2001100
    \(P\)- 1- 2- 50000
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)

  3. 6. (a) Value of initial flow
    (b) Capacity of cut \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d5798c81-290a-4e4b-aa46-497b62ca899b-20_1931_1099_507_408} \captionsetup{labelformat=empty} \caption{Diagram 1}
    \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{d5798c81-290a-4e4b-aa46-497b62ca899b-21_1874_953_653_589}
    7. You may not need to use all the rows of this table
    StageStateActionDest.Value
    T-shirt
    StageStateActionDest.Value
    Maximum annual profit £ \(\_\_\_\_\)
Edexcel D2 2018 June Q1
7 marks Moderate -0.5
  1. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of four supply points, \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , to each of four demand points, \(1,2,3\) and 4 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution to this transportation problem is required.
\begin{table}[h]
1234Supply
A2432213427
B2831293741
C2541333531
D2332313614
Demand33352520
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows an initial solution given by the north-west corner method. \begin{table}[h]
1234
A27
B635
C0256
D14
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Explain why a zero has been placed in cell C 2 in Table 2. State the other cell in Table 2 in which the zero could have been placed.
  2. State the shadow costs clearly and enter the improvement indices into Table 3 in your answer book. Taking the most negative improvement index to indicate the entering cell,
    [0pt]
  3. list the stepping-stone route that should be used to obtain the next solution. You should make clear the cells that are included in your route and state your entering and exiting cells. [You do not need to state the next solution. You do not need to solve this problem.]
Edexcel D2 2018 June Q2
13 marks Moderate -0.8
2. A two-person zero-sum game is represented by the following pay-off matrix for player A.
B plays 1B plays 2B plays 3B plays 4
A plays 1-325-1
A plays 2-531-1
A plays 3-2542
A plays 42-3-14
  1. Identify the play safe strategies for each player.
  2. State, giving a reason, whether there is a stable solution to this game.
  3. Explain why the game above can be reduced to the following \(3 \times 3\) game.
    - 325
    - 254
    2- 3- 1
  4. Formulate the \(3 \times 3\) game as a linear programming problem for player A, defining your variables clearly and writing the constraints as inequalities.
Edexcel D2 2018 June Q3
11 marks Standard +0.3
3. Five workers, A, B, C, D and E, are to be assigned to five tasks, 1, 2, 3, 4 and 5 . Each worker must be assigned to only one task and each task must be done by only one worker. The cost, in pounds, of assigning each worker to each task is shown in the table below. The cost of assigning worker D to task 4 is \(\pounds x\), where \(x > 38\)
12345
A2531272935
B2933403537
C2829353637
D343536\(x\)41
E3635323133
The total cost is to be minimised.
  1. Reducing rows first, use the Hungarian algorithm to obtain an allocation that minimises the cost. You must make your method clear and show the table after each stage.
    (8)
  2. Find the minimum total cost.
    (1) Workers A and D decide that they do not like the task they have been allocated and are allowed to swap tasks with each other. The other three allocations are unchanged. The cost now of allocating the five workers to the five tasks is now \(\pounds 5\) more than the minimum cost found in (b).
  3. Calculate the value of \(x\). You must show your working.
    (2)
Edexcel D2 2018 June Q4
12 marks Moderate -0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4abb2325-b9df-4849-b08c-7db465fe85e0-05_1054_1569_194_248} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 represents a system of pipes through which fluid can flow from the source node, S , to the sink node, T. The labelling procedure has been applied to Figure 1, and the numbers on the arrows, either side of each arc, show the excess capacities and potential backflows. Currently, no fluid is flowing through the system.
  1. Calculate the capacity of the cut that passes through arcs \(\mathrm { GT } , \mathrm { EG } , \mathrm { DE } , \mathrm { BE } , \mathrm { FE }\) and FH .
  2. Explain why arc GT can never be full to capacity when fluid is flowing through the system.
  3. Apply the labelling procedure to Diagram 1 in the answer book to show the maximum flow along SBET. State the amount that can flow along this route.
  4. Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  5. State the maximum flow through the system and find a cut to show that this flow is maximal.
  6. Show the maximum flow on Diagram 2 in the answer book.
Edexcel D2 2018 June Q5
17 marks Challenging +1.2
5. The initial tableau for a linear programming problem in \(x , y\) and \(z\) is shown below. The objective function to be maximised is \(P = 4 x + 2 y + k z\), where \(k\) is a positive constant.
Basic Variable\(x\)\(y\)\(z\)r\(s\)\(t\)Value
\(r\)-2-6110040
\(s\)23201080
\(t\)12200150
\(P\)-4-2-k0000
  1. Using the information in the tableau, write down the three constraints as inequalities.
  2. By increasing \(x\), perform one complete iteration of the simplex algorithm to obtain tableau \(T _ { 1 }\) and state the row operations you use.
  3. Given that \(T _ { 1 }\) is not optimal, find an inequality for the value of \(k\).
  4. Perform a second complete iteration of the simplex algorithm to obtain tableau \(T _ { 2 }\) and state the row operations you use.
  5. Given that \(T _ { 2 }\) is optimal, find a second inequality for the value of \(k\).
  6. State the final value of each variable and give an expression for the final value of \(P\) in terms of \(k\).
  7. Hence find the range of possible values of \(P\).
Edexcel D2 2018 June Q6
15 marks Standard +0.3
6. Jonathan is an author who is planning his next book tour. He will visit four countries over a period of four weeks. He will visit just one country each week. He will leave from his home, S , and will only return there after visiting the four countries. He will travel directly from one country to the next. He wishes to determine a schedule of four countries to visit. Table 1 shows the countries he could visit each week. \begin{table}[h]
1234
A
B
C
D
1234
A
B
C
D
1234
A
B
C
D
2.
B plays 1B plays 2B plays 3B plays 4
A plays 1- 325- 1
A plays 2- 531- 1
A plays 3- 2542
A plays 42- 3- 14
- 325
- 254
2- 3- 1
3.
12345
A2531272935
B2933403537
C2829353637
D343536\(x\)41
E3635323133
  1. You may not need to use all of these tables
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    12345
    A
    B
    C
    D
    E
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4abb2325-b9df-4849-b08c-7db465fe85e0-18_1056_1572_1450_185} \captionsetup{labelformat=empty} \caption{Diagram 1}
    \end{figure} Maximum flow along SBET: \(\_\_\_\_\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4abb2325-b9df-4849-b08c-7db465fe85e0-19_1043_1572_1505_187} \captionsetup{labelformat=empty} \caption{Diagram 2}
    \end{figure} 5.
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)- 2- 6110040
    \(s\)23201080
    \(t\)12200150
    \(P\)- 4- 2\(- k\)0000
    You may not need to use all of these tableaux
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
    \(P\)
    6.
    StageStateActionDestinationValue
    0IISS\(30 - 5 = 25 ^ { * }\)
    StageStateActionDestinationValue
    END
Edexcel D2 2019 June Q1
5 marks Moderate -0.8
1.
\cline { 2 - 7 } \multicolumn{1}{c|}{}ABCDEF
A-5347393540
B53-32464143
C4732-514737
D394651-3649
E35414736-42
F4043374942-
The table above shows the least distances, in km, between six towns, A, B, C, D, E and F. Jas needs to visit each town, starting and finishing at D , and wishes to minimise the total distance she travels.
  1. Starting at D , use the nearest neighbour algorithm to obtain an upper bound for the length of the route. You must state your route and its length.
  2. Starting by deleting D , and all of its arcs, find a lower bound for the route length.
Edexcel D2 2019 June Q2
10 marks Moderate -0.3
2. Table 1 shows the cost, in pounds, of transporting one unit of stock from each of three supply points, \(\mathrm { A } , \mathrm { B }\) and C , to each of four demand points, 1, 2, 3 and 4 . It also shows the stock held at each supply point and the stock required at each demand point. A minimum cost solution is required. \begin{table}[h]
1234Supply
A1720231425
B1615192229
C1914111532
Demand28172318
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} Table 2 shows an initial solution given by the north-west corner method. \begin{table}[h]
1234
\(A\)25
\(B\)3179
\(C\)1418
\captionsetup{labelformat=empty} \caption{Table 2}
\end{table}
  1. Taking A4 as the entering cell, use the stepping-stone method to find an improved solution. Make your route clear.
  2. Taking the most negative improvement index to indicate the entering cell, use the stepping-stone method once to obtain an improved solution. You must make your method clear by stating your shadow costs, improvement indices, route, entering cell and exiting cell.
  3. Determine whether your current solution is optimal, giving a reason for your answer.
  4. State the cost of your current solution.
Edexcel D2 2019 June Q3
8 marks Moderate -0.8
3. Five friends have rented a house that has five bedrooms. They each require their own bedroom. The table below shows how each friend rated the five bedrooms, \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E , where 0 is low and 10 is high.
ABCDE
Frank50734
Gill538101
Harry43790
Imogen63654
Jiao02732
Reducing rows first, use the Hungarian algorithm to obtain an allocation that maximises the total of all the ratings. You must make your method clear and show the table after each stage.
(Total 8 marks)
Edexcel D2 2019 June Q4
12 marks Standard +0.3
4. Eugene and Stephen play a zero-sum game. The pay-off matrix shows the number of points that Eugene scores for each combination of strategies.
Stephen plays 1Stephen plays 2Stephen plays 3
Eugene plays 1450
Eugene plays 2-211
Eugene plays 3-3-43
  1. Find the play-safe strategies for each of Eugene and Stephen, and hence show that this zero-sum game does not have a stable solution.
  2. Suppose that Eugene knows that Stephen will use his play-safe strategy. Explain why Eugene should change from his play-safe strategy. You should state as part of your answer which strategy Eugene should now play.
  3. Formulate the game as a linear programming problem for Stephen. Define your variables clearly. Write the constraints as equations.
Edexcel D2 2019 June Q5
11 marks Standard +0.3
5. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(P = 2 x + 3 y + z\) subject to \(\quad 2 y - 3 z \leqslant 30\) $$\begin{array} { r } - 3 x + y + z \leqslant 60 \\ x + 4 y - z \leqslant 80 \end{array}$$
  1. Complete the initial tableau in the answer book for this linear programming problem.
    (3)
  2. Taking the most negative number in the profit row to indicate the pivot column, perform one complete iteration of the simplex algorithm to obtain a new tableau, T. Make your method clear by stating the row operations you use.
    (5)
  3. Write down the profit equation given by T and state the values of the slack variables given by T . The following tableau is obtained after further iterations.
    Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)02-310030
    \(s\)013-2013300
    \(x\)14-100180
    \(P\)05-3002160
  4. Explain why no optimal solution can be found by applying the simplex algorithm to the above tableau.
Edexcel D2 2019 June Q6
14 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0c66144-9e34-42fc-9f40-a87a49331483-07_719_1313_246_376} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow.
  1. State the value of the initial flow.
    1. Add a supersource, S , and a supersink, T , and corresponding arcs to Diagrams 1 and 2 in the answer book.
    2. Enter the flow value and appropriate capacity on each of the arcs you have added to Diagram 1.
  3. Complete the initialisation of the labelling procedure on Diagram 2 in the answer book by entering values along the new arcs from S to T , and along \(\operatorname { arcs } \mathrm { S } _ { 1 } \mathrm {~B}\) and \(\mathrm { AT } _ { 1 }\)
  4. Hence use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
  5. Draw a maximal flow pattern on Diagram 3 in the answer book.
  6. Prove that your flow is maximal.
Edexcel D2 2019 June Q7
15 marks Moderate -0.5
7. A company has purchased a plot of land and has decided to build four holiday homes, A, B, C and D, on the land at the rate of one home per year. The company expects that the construction costs each year will vary, depending on which houses have already been constructed and which house is currently under construction. The expected construction costs, in thousands of pounds, are shown in the table below.
\cline { 2 - 7 } \multicolumn{1}{c|}{}ABCDEF
A-5347393540
B53-32464143
C4732-514737
D394651-3649
E35414736-42
F4043374942-
\begin{table}[h]
1234Supply
A1720231425
B1615192229
C1914111532
Demand28172318
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table} 2. You may not need to use all of these tables \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Table 1}
1234Supply
A25
B29
C32
Demand28172318
\end{table}
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
1234Supply
A25
B29
C32
Demand28172318
3.
ABCDE
Frank50734
Gill538101
Harry43790
Imogen63654
Jiao02732
You may not need to use all of these tables
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
ABCDE
F
G
H
I
J
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
\(A\)\(B\)\(C\)\(D\)\(E\)
\(F\)
\(G\)
\(H\)
\(I\)
\(J\)
Stephen plays 1Stephen plays 2Stephen plays 3
Eugene plays 1450
Eugene plays 2-211
Eugene plays 3-3-43
4. 5. (a)
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
30
60
80
0
You may not need to use all of these tableaux
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
\(P\)
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
\(P\)
b.v.\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)ValueRow Ops
\(P\)
6. (a) Value of initial flow
(b) and (c) \includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-20_725_1251_404_349} \section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-20_1070_1264_1322_349}
\section*{Diagram 2} (d)
(e) \includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-21_714_1385_1306_283} \section*{Diagram 3} (f)
7. (a)
StageStateActionDest.Value
1ABCDABCD65*
StageStateActionDest.Value
\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-24_2642_1833_118_118}
OCR D2 2006 June Q1
14 marks Standard +0.3
1 The network represents a system of pipes along which fluid can flow from \(S\) to \(T\). The values on the arcs are lower and upper capacities in litres per second. \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-02_696_1292_376_424}
  1. Calculate the capacity of the cut with \(\mathrm { X } = \{ S , A , B , C \} , \mathrm { Y } = \{ D , E , F , G , H , I , T \}\).
  2. Show that the capacity of the cut \(\alpha\), shown on the diagram, is 12 litres per second and calculate the minimum flow across the cut \(\alpha\), from \(S\) to \(T\), (without regard to the remainder of the diagram).
  3. Explain why the arc SC must have at least 5 litres per second flowing through it. By considering the flow through \(A\), explain why \(A D\) cannot be full to capacity.
  4. Show that it is possible for 11 litres per second to flow through the system.
  5. From your previous answers, what can be deduced about the maximum flow through the system?
OCR D2 2006 June Q2
15 marks Moderate -0.3
2 A delivery company needs to transport heavy loads from its warehouse to a ferry port. Each of the roads that it can use has a bridge with a maximum weight limit. The directed network below represents the roads that can be used to get from the warehouse to the ferry port. Road junctions are labelled with (stage; state) labels. The weights on the arcs represent weight limits in tonnes. \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-03_896_1561_468_292}
  1. Explain what a maximin route is.
  2. Set up a dynamic programming tabulation, working backwards from stage 1, to find the two maximin routes through the network. What is the maximum load that can be transported in one journey from the warehouse to the ferry port?
  3. A new road is opened connecting ( \(2 ; 0\) ) and ( \(2 ; 1\) ). There is no bridge on this road so it does not restrict the weight of the load that can be carried. Using the new road, what is the maximum load that can be transported in one journey from the warehouse to the ferry port? State the route that the delivery company should use. (Do not attempt to apply dynamic programming in this part.)
OCR D2 2006 June Q3
14 marks Standard +0.3
3 Rose and Colin repeatedly play a zero-sum game. The pay-off matrix shows the number of points won by Rose for each combination of strategies.
\multirow{6}{*}{Rose's strategy}Colin's strategy
\(W\)\(X\)\(Y\)\(Z\)
\(A\)-14-32
\(B\)5-256
C3-4-10
\(D\)-56-4-2
  1. What is the greatest number of points that Colin can win when Rose plays strategy \(A\) and which strategy does Colin need to play to achieve this?
  2. Show that strategy \(B\) dominates strategy \(C\) and also that strategy \(Y\) dominates strategy \(Z\). Hence reduce the game to a \(3 \times 3\) pay-off matrix.
  3. Find the play-safe strategy for each player on the reduced game. Is the game stable? Rose makes a random choice between the strategies, choosing strategy \(A\) with probability \(p _ { 1 }\), strategy \(B\) with probability \(p _ { 2 }\) and strategy \(D\) with probability \(p _ { 3 }\). She formulates the following LP problem to be solved using the Simplex algorithm: $$\begin{array} { l l } \text { maximise } & M = m - 5 , \\ \text { subject to } & m \leqslant 4 p _ { 1 } + 10 p _ { 2 } , \\ & m \leqslant 9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 } , \\ & m \leqslant 2 p _ { 1 } + 10 p _ { 2 } + p _ { 3 } , \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 , \\ \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$ (You are not required to solve this problem.)
  4. Explain how \(9 p _ { 1 } + 3 p _ { 2 } + 11 p _ { 3 }\) was obtained. A computer gives the solution to the LP problem as \(p _ { 1 } = \frac { 7 } { 48 } , p _ { 2 } = \frac { 27 } { 48 } , p _ { 3 } = \frac { 14 } { 48 }\).
  5. Calculate the value of \(M\) at this solution.
OCR D2 2006 June Q4
14 marks Moderate -0.5
4 Answer this question on the insert provided. The diagram shows an activity network for a project. The table lists the durations of the activities (in hours). \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-05_680_1125_424_244} (ii) Key: \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_154_225_1119_1509} \captionsetup{labelformat=empty} \caption{Early event Late event time time}
\end{figure}
\includegraphics[max width=\textwidth, alt={}]{e879b1f5-edc7-4819-80be-2a90dbf3d451-10_762_1371_1409_427}
Minimum completion time = \(\_\_\_\_\) hours Critical activities: \(\_\_\_\_\) (iii) \(\_\_\_\_\) (iv) \includegraphics[max width=\textwidth, alt={}, center]{e879b1f5-edc7-4819-80be-2a90dbf3d451-11_513_1189_543_520} Number of workers required = \(\_\_\_\_\)
(i)\(A \bullet\)
\(B \bullet\)\(\bullet J\)
\(C \bullet\)\(\bullet K\)
\(D \bullet\)\(\bullet L\)
\(E \bullet\)\(\bullet M\)
\(F \bullet\)\(\bullet N\)
(ii) \(\_\_\_\_\) (iii)
\(J\)\(K\)\(L\)\(M\)\(N\)\(O\)
\(A\)252252
\(B\)252055
\(C\)505522
\(D\)
\(E\)
\(F\)
Answer part (iv) in your answer booklet.
OCR D2 2010 June Q1
6 marks Moderate -0.8
1 The famous fictional detective Agatha Parrot is investigating a murder. She has identified six suspects: Mrs Lemon \(( L )\), Prof Mulberry \(( M )\), Mr Nutmeg \(( N )\), Miss Olive \(( O )\), Capt Peach \(( P )\) and Rev Quince \(( Q )\). The table shows the weapons that could have been used by each suspect.
Suspect
\(L\)M\(N\)\(O\)\(P\)\(Q\)
Axe handleA✓✓✓
Broomstick\(B\)✓✓
DrainpipeD✓✓
Fence post\(F\)✓✓
Golf club\(G\)✓✓✓
Hammer\(H\)✓✓✓
  1. Draw a bipartite graph to represent this information. Put the weapons on the left-hand side and the suspects on the right-hand side. Agatha Parrot is convinced that all six suspects were involved, and that each used a different weapon. She originally thinks that the axe handle was used by Prof Mulberry, the broomstick by Miss Olive, the drainpipe by Mrs Lemon, the fence post by Mr Nutmeg and the golf club by Capt Peach. However, this would leave the hammer for Rev Quince, which is not a possible pairing.
  2. Draw a second bipartite graph to show this incomplete matching.
  3. Construct the shortest possible alternating path from \(H\) to \(Q\) and hence find a complete matching. Write down which suspect used each weapon.
  4. Find a different complete matching in which none of the suspects used the same weapon as in the matching from part (iii).
OCR D2 2010 June Q2
10 marks Moderate -0.5
2 In an investigation into a burglary, Agatha has five suspects who were all known to have been near the scene of the crime, each at a different time of the day. She collects evidence from witnesses and draws up a table showing the number of witnesses claiming sight of each suspect near the scene of the crime at each possible time. Suspect \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Time}
1 pm2 pm3 pm4 pm5 pm
Mrs Rowan\(R\)34271
Dr Silverbirch\(S\)510666
Mr Thorn\(T\)47353
Ms Willow\(W\)68483
Sgt Yew\(Y\)88743
\end{table}
  1. Use the Hungarian algorithm on a suitably modified table, reducing rows first, to find the matchings for which the total number of claimed sightings is maximised. Show your working clearly. Write down the resulting matchings between the suspects and the times. Further enquiries show that the burglary took place at 5 pm , and that Dr Silverbirch was not the burglar.
  2. Who should Agatha suspect?